The Minchas Chinuch writes in Mitzva שי"ג that since there is a prohibition to feed a child an issur it is prohibited to feed children on Yom Kippur any more then they need. He states that children between 3 and 8, you should not feed at all. It comes out from the Minchas Chinuch that for children who can feed themselves it should be prohibited for you to feed them and in fact even a baby you can get a non-Jew to feed. (also see the Mishna Berura סימן תרט"ז ס"ק ה)
This Minchas Chinuch is clearly not accepted l'halacha the question is why not?
R' Tzvi Pesach Frank in Mikraie Kodesh as well as other Acharonim offer the following suggestion.
By other issurim (feeding a child pig) the food has a שם איסור, it is an איסור תפצא on the food and therefore even though the child is not chayav in mitzvos we are prohibited to feed him issurim, it is like feeding him poison. However, food on Yom Kippur is clearly an issur on the person, the kosher food does not suddenly turn into an issur, the person is prohibited from eating on Yom Kippur. Therefore, a child who has no issur of eating, for them the food is completely permitted and they can eat it. Since they are allowed to eat it the parents are allowed to feed them.